Find the gradient of $f(x, y, z) = z^\pi - xy$. $\nabla f = ($ $,$ $,$ $)$
The gradient of a scalar field is all its partial derivatives put together into a vector. For a 3D scalar field, this looks like $\nabla f = (f_x, f_y, f_z)$. Let's find $f_x$, $f_y$, and $f_z$. $\begin{aligned} f_x &= \dfrac{\partial}{\partial x} \left[ z^\pi - xy \right] \\ \\ &= -y \\ \\ f_y &= \dfrac{\partial}{\partial y} \left[ z^\pi - xy \right] \\ \\ &= -x \\ \\ f_z &= \dfrac{\partial}{\partial z} \left[ z^\pi - xy \right] \\ \\ &= \pi z^{\pi -1} \end{aligned}$ The gradient of $f$ is: $\nabla f = \left( -y, -x, \pi z^{\pi - 1} \right)$